The wave equation is

(1) 
where is the Laplacian.
The 1D wave equation is

(2) 
In order to specify a wave, the equation is subject to boundary conditions

(3) 

(4) 
and initial conditions
The wave equation can be solved using the socalled d'Alembert's solution, a Fourier Transform method, or
Separation of Variables.
d'Alembert devised his solution in 1746, and Euler subsequently expanded the method in 1748. Let

(7) 

(8) 
By the Chain Rule,

(9) 

(10) 
The wave equation then becomes

(11) 
Any solution of this equation is of the form

(12) 
where and are any functions. They represent two waveforms traveling in opposite directions, in the
Negative direction and in the Positive direction.
The 1D wave equation can also be solved by applying a Fourier Transform to each side,

(13) 
which is given, with the help of the Fourier Transform Derivative identity, by

(14) 
where

(15) 
This has solution

(16) 
Taking the inverse Fourier Transform gives
where

(18) 

(19) 
This solution is still subject to all other initial and boundary conditions.
The 1D wave equation can be solved by Separation of Variables using a trial solution

(20) 
This gives

(21) 

(22) 
So the solution for is

(23) 
Rewriting (22) gives

(24) 
so the solution for is

(25) 
where
. Applying the boundary conditions
to (23) gives

(26) 
where is an Integer. Plugging (23), (25) and (26) back in for in (21) gives, for a
particular value of ,
The initial condition
then gives , so (27) becomes

(28) 
The general solution is a sum over all possible values of , so

(29) 
Using Orthogonality of sines again,

(30) 
where is the Kronecker Delta defined by

(31) 
gives
so we have

(33) 
The computation of s for specific initial distortions is derived in the Fourier Sine Series section. We already have found
that , so the equation of motion for the string (29), with

(34) 
is

(35) 
where the Coefficients are given by (33).
A damped 1D wave

(36) 
given boundary conditions
initial conditions
and the additional constraint

(41) 
can also be solved as a Fourier Series.

(42) 
where
To find the motion of a rectangular membrane with sides of length and (in the absence of
gravity ), use the 2D wave equation

(46) 
where is the vertical displacement of a point on the membrane at position () and time . Use
Separation of Variables to look for solutions of the form

(47) 
Plugging (47) into (46) gives

(48) 
where the partial derivatives have now become complete derivatives. Multiplying (48) by gives

(49) 
The left and right sides must both be equal to a constant, so we can separate the equation by writing the right side as

(50) 
This has solution

(51) 
Plugging (50) back into (49),

(52) 
which we can rewrite as

(53) 
since the left and right sides again must both be equal to a constant. We can now separate out the equation

(54) 
where we have defined a new constant satisfying

(55) 
Equations (53) and (54) have solutions

(56) 

(57) 
We now apply the boundary conditions to (56) and (57). The conditions and mean that

(58) 
Similarly, the conditions and give
and
, so
and , where and are Integers. Solving for the allowed values of and then gives

(59) 
Plugging (52), (56), (57), (58), and (59) back into (22) gives the solution for particular
values of and ,

(60) 
Lumping the constants together by writing
(we can do this since is a function of
and , so can be written as ) and
, we obtain

(61) 
Plots of the spatial part for modes (1, 1), (1, 2), (2, 1), and (2, 2) follow.
The general solution is a sum over all possible values of and , so the final solution is

(62) 
where is defined by combining (55) and (59) to yield

(63) 
Given the initial conditions and
, we can compute the s and
s explicitly. To accomplish this, we make use of the orthogonality of the Sine function in the form

(64) 
where is the Kronecker Delta.
This can be demonstrated by direct Integration. Let
so
in (64), then

(65) 
Now use the trigonometric identity

(66) 
to write

(67) 
Note that for an Integer , the following Integral vanishes
since when is an Integer. Therefore, when
. However, does not
vanish when , since

(69) 
We therefore have that
, so we have derived (64). Now we multiply by two sine
terms and integrate between 0 and and between 0 and ,

(70) 
Now plug in , set , and prime the indices to distinguish them from the and in (70),
Making use of (64) in (71),

(72) 
so the sums over and collapse to a single term

(73) 
Equating (72) and (73) and solving for then gives

(74) 
An analogous derivation gives the s as

(75) 
The equation of motion for a membrane shaped as a Right Isosceles Triangle of length
on a side and with the sides oriented along the Positive and axes is given by





(76) 
where

(77) 
and , Integers with . This solution can be obtained by subtracting two wave solutions for a
square membrane with the indices reversed. Since points on the diagonal which are equidistant from the center must have the
same wave equation solution (by symmetry), this procedure gives a wavefunction which will vanish along the diagonal as
long as and are both Even or Odd. We must further restrict the modes since those with give
wavefunctions which are just the Negative of and give an identically zero wavefunction. The following
plots show (3, 1), (4, 2), (5, 1), and (5,3).
References
Abramowitz, M. and Stegun, C. A. (Eds.). ``Wave Equation in Prolate and Oblate Spheroidal Coordinates.'' §21.5 in
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 752753, 1972.
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGrawHill, pp. 124125, 1953.
© 19969 Eric W. Weisstein
19990526